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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 391
Next de�ne θ = πx/L so that the limits are θ = π/2 to θ = π, and dx = (L/π)dθ
= −(2/L)(L/π)∫
π
π/2
sin θ cos θ dθ
�e integral is evaluated by noting that d/dθ(sin2 θ) = 2 sin θ cos θ
= −(2/L)(L/π) 12 ∣sin
2 θ∣ππ/2 = −(2/L)(L/π) 12 (0 − 1) = 1/π
�e Franck–Condon factor is I2 = 1/π2 ; numerically this is 0.318.
E11F.7(b) �e wavenumbers of the lines in the R branch are given in [11F.7–465], ν̃R(J) =
ν̃ + (B̃′ + B̃)(J + 1) + (B̃′ − B̃)(J + 1)2. �e band head is located by �nding
the value of J which gives the largest wavenumber, which can be inferred by
solving dν̃R(J)/dJ = 0.
d
dJ
[ν̃ + (B̃′ + B̃)(J + 1) + (B̃′ − B̃)(J + 1)2] = (B̃′ + B̃) + 2(J + 1)(B̃′ − B̃)
Setting the derivative to zero and solving for J gives
Jhead =
−(B̃′ + B̃)
2(B̃′ − B̃)
− 1 = B̃ − 3B̃′
2(B̃′ − B̃)
A band head only occurs in the R branch if B̃′ B̃ a band head will occur in the P branch.
�e wavenumbers of the lines in the P branch are given in [11F.7–465], ν̃P(J) =
ν̃ − (B̃′ + B̃)J + (B̃′ − B̃)J2. �e band head is located by �nding the value
of J which gives the smallest wavenumber, which can be inferred by solving
dν̃P(J)/dJ = 0.
d
dJ
[ν̃ − (B̃′ + B̃)J + (B̃′ − B̃)J2] = −(B̃′ + B̃) + 2J(B̃′ − B̃)
Setting the derivative to zero and solving for J gives
Jhead =
B̃′ + B̃
2(B̃′ − B̃)
With the data given
Jhead =
10.470 + 10.308
2(10.470 − 10.308)
= 64.1
Assuming that it is satisfactory simply to round this to the nearest integer the
band head occurs at J = 64 .